Embedding Degree of Hyperelliptic Curves with Complex Multiplication

TL;DR

This paper investigates the embedding degree of Jacobians of genus two hyperelliptic curves with complex multiplication over finite fields, revealing a specific condition under which the embedding degree is one.

Contribution

It establishes a new criterion linking the structure of the l-Sylow subgroup to the embedding degree of Jacobians with complex multiplication.

Findings

If the l-Sylow subgroup is not cyclic, the embedding degree is one.

Provides insight into the relationship between subgroup structure and cryptographic properties.

Enhances understanding of hyperelliptic curve Jacobians in finite field cryptography.

Abstract

Consider the Jacobian of a genus two curve defined over a finite field and with complex multiplication. In this paper we show that if the l-Sylow subgroup of the Jacobian is not cyclic, then the embedding degree of the Jacobian with respect to l is one.